Trotter error with commutator scaling for the Fermi-Hubbard model

TL;DR

This paper develops refined error bounds for Trotter product formulas applied to the Fermi-Hubbard model, providing explicit expressions based on lattice geometry and Hamiltonian parameters, with implications for quantum simulation accuracy.

Contribution

It generalizes existing Trotter error bounds to higher orders with smaller prefactors and applies them to complex lattice geometries in the Fermi-Hubbard model.

Findings

Derived explicit error bounds for 1D and 2D lattices

Bound overestimation compared to actual errors observed

Provided symbolic evaluation methods for nested commutators

Abstract

We derive higher-order error bounds with small prefactors for a general Trotter product formula, generalizing a result of Childs et al. [Phys. Rev. X 11, 011020 (2021)]. We then apply these bounds to the real-time quantum time evolution operator governed by the Fermi-Hubbard Hamiltonian on one-dimensional and two-dimensional square and triangular lattices. The main technical contribution of our work is a symbolic evaluation of nested commutators between hopping and interaction terms for a given lattice geometry. The calculations result in explicit expressions for the error bounds in terms of the time step and Hamiltonian coefficients. Comparison with the actual Trotter error (evaluated on a small system) indicates that the bounds still overestimate the error.